October 23, 2006

Solving Problems Makes Your Brain Hurt

(This is part five of a multi-part series on your baiin and why constructivism doesn't work. Start at part one.)

In part four, we learned how knowing a lot of stuff helps you learn, think about, and retain new information and skills. In this installment we'll learn why learning by solving problems, the main theory behind most of constructivism, frequently makes learning more difficult.

The theory underlying constructivism is that people learn best in an unguided or minimimally guided environment , generally defined as one in which students, rather than being persented with essential information, must discover or construct essential information for themeselves. Thus, constructivism requires the novice student to search a problem space for problem-relevant information to discover the essential information, i.e., the solution to the problem posed.

As we learned in previous installments, this search must occur in our limited working memory (WM). All problem-based searching makes heavy demands on WM and often overburdens the limited WM, requiring WM resources be used for activities that are unrelated to learning. Therefore, while your brain's WM is consumed searching for a problem solution, it's not contributing to the accumulation of knowledge in long-term memory (LTM). As a consequence, students can engage in problem-solving activities for extended periods and learn almost nothing (i.e., store what's been learned in LTM).

Bear in mind, the goal of instruction is rarely just to search for or discover information. The goal of education is to give learners specific guidance about how to cognitively manipulate information in ways that are consistent with a learning goal, and store the result in LTM.

Allegedly, constructivism helps students to derive meaning from learning materials. However, cognitive load theory suggests that the free exploration of a highly complex environment generates a heavy WM load that is detrimental to learning. This suggestion is particularly important in the case of novice learners who lack proper cogntive LTM structures to integrate the new information with their prior knowledge.

The consequences of requiring novice students to search for problem solutions using a limited WM or the mechanisms by which unguided or minimally guided instruction might facilitate change in LTM appear to be routinely ignored in constructivist theory. The result is a set of differently named but similar instructional approaches requiring minimal guidance that are disconnected from much that we know of human cognition.

Novice students learn much more from examining worked problem examples rather than solving the problems themselves. This effect only disappears and reverses as the student's expertise increases. Problem solving only becomes relatively effective when learners are sufficiently experienced so that studying a worked example is, for them, a redundant activity that increases WM load compared to generating a known solution. This phenomenon emphasizes the importance of providing novices in an area with extensive guidance because they do not have sufficient knowledge in LTM to prevent unproductive problem-solving search. That guidance can be relaxed only with increased expertise as knowledge in LTM can take over from external guidance. The vast majority of students at the K-12 level are novices in the subjects they are learning who do not have the needed expertise.

Because novice students learn so little from the constructivist approach, most teachers who attempt to implement classroom-based constructivist instruction end up providing students with considerable guidance or direct instruction. In fact, teachers frequently provide a lot of “scaffolding” when students fail to make learning progress in a discovery setting. If they expect to be effective, teachers typically spend a great deal of time in instructional interactions with students by:

simultaneously teaching content and scaffolding-relevant procedures … by (a) modeling procedures for identifying and self-checking important information…(b) showing students how to reduce that information to paraphrases … (c) having students use notes to construct collaborations and routines, and (d) promoting collaborative dialogue within problems.

One of the more frequent excuses for retaining the inquiry or problem solving instructional theme, even when it is heavily scaffolded with direct instruction and guidance, is the belief that providing students with partial information enhances their ability to construct an LTM representation more than giving them full information. This is incorrect.

Most students of all ages know how to construct knowledge when given adequate information and there is no evidence that presenting them with partial information is beneficial. Actually, the resevesr seems most often to be true. Students must construct a mental representation or schema in LTM regardless of whether they are given partial or complete information. Complete information will result in a more accurate representation that is also more easily acquired.

And, while there is little doubt that students remember material they generate themselves better than material that is handed to them. This "generation effect," as it is called, is indeed powerful, and it is due, in part, to forcing the learner to think about the meaning of material (although other techniques can do that as well). Part of the effect does seem to be unique to the actual generation of the answer, over and above thinking about meaning. For this reason constructivists believe that discovery learning should be employed whenever possible. However, given that memory follows thought, one thing is clear: Students will remember incorrect "discoveries" just as well as correct ones.

Altering LTM is the ultimate goal of learning. The generation effect does affect how information is learned, but so do many other factors that we've discussed in previous installments. The main underlying flaw of constructvism is the undue attention paid to the generation effect to the detriment of recognizing the important limitations of our working memory. Moreover, it ignores the very real negative byproducts when students acquire misconceptions or incomplete or disorganized knowledge. Today, there is little evidence that supports the notion that learning under a constructivist pedagogy is as good as, much less improved, over well designed direct instruction at the K-12 level.

This brings us to the point where we can finally discuss the remaining underlying fallacies of constructivism. We'll do that in the next installment.

13 comments:

Anonymous
said...

Thank you for your post. I truly have learned a great deal from reading your blog. My son is in the third grade and the math curriculum is based on "dicovery learning." Will he learn anything this year? He is weak on basic math facts...something he should have mastered in the second grade. I have to un-school him every day when he comes home. It is exhausting. Can you offer a mom any ray of hope?Sincerly,Worried in Virginia

Don't make it more difficult than it has to be. I wouldn't say that I have to "un-school" my Everyday Math son. I just make sure that he masters the basics. If you want, you can get Singapore Math or Saxon Math books and follow them carefully. However, In third grade, things are not too bad yet. Make sure he knows his times table and masters(!) the traditional algorithms for add, subtract, multiply, and divide (4th grade perhaps). There are places on-line where you can print out worksheets - Google for something like "math worksheets". These worksheets will get you started while you look into what you can do over the long term (Singapore, Saxon, or Kumon, if they are in your area.) Math is not difficult if you make sure your son masters the basics. This is perhaps difficult when your son still has to do the fuzzy homework, but mastery of the basic skills is mandatory. Schools don't do this because they mix up mastery with "rote".

I'll second Steve's comments. Just make sure your son has discovered any misconceptions. Clear those up first, then just give sufficient practice in the basics until he can answer quickly without fumbling. He shouldn't have to think when applying the standard algorithms.

Hmmm, I'm not sure how well discovered-solutions go into LTM. I can think of a fair number of times in my working life where I've managed to come up with a solution to a problem, but I now can't recall what the solution was. I can develop an algorithm, test it, implement it if it works, and then forget all the details if I don't practice them.

I feel I haven't made it more difficult than it has to be..the teacher has. The class work consists of very little basic math work...a few problems daily and the homework is non-existent. The teacher and the principal do not believe in "drill or kill"(their words not mine). The principal and the teacher are evasive when questioned about the third grade math curriculum. The principal is a huge fan of Alfie Kohn.

Also, I wonder is this something I am going to have to deal with year after year. Yes, he is only in third grade, but what about fourth grade and so on and so on. I have read about Everyday Math and it doesn't seem like a good program. So our kids have these bad math programs and they come home to be tutored or go someplace else to be tutored? By the way, I have a friend whose kid is failing Everyday Math. He is in Pre-AP classes and had done well up until this program was introduced. That is crappy!

"Also, I wonder is this something I am going to have to deal with year after year."

Yes. Join the club. Private schools provide no guarantees either. My son is in a private school (fifth grade - Everyday Math) and some kids don't know what 6 times 7 is. I want the teacher/school to go, go, go, but they can't/won't. We use Singapore Math at home.

The key is getting your son into a proper algebra course in 8th grade. Some larger schools have enough students to offer "advanced" algebra courses in 8th grade. The school won't provide a proper path to this course, so the kids in this course are either very natural at math, or they got outside help. Don't assume that if a school has a good algebra course, they provide a proper path to get there. Anything less than the top algebra course in 8th grade will put your son on a lower track in high school.

In our town, many kids (with good grades) and their parents are taken completely by surprise when they get to high school. High schools usually do a better job of providing proper courses - at least in the AP tracks. You just can't rely on the lower schools to prepare your kids to get to these tracks, even if your child has all A's.

"The teacher and the principal do not believe in "drill or kill"(their words not mine)."

This means that they don't believe in mastery. Kids will enter the sixth grade not knowing their times table. When everything falls apart in 8th grade, the kids will say either that they just don't like math or they are not "math brains". Enough kids will do well (in spite of the school or because of outside tutoring) that nobody will think that there is a problem.

"So our kids have these bad math programs and they come home to be tutored or go someplace else to be tutored?"

Sad, but true.

"He is in Pre-AP classes ..."

I'm not sure what this is, but Everyday Math just goes through 6th grade. I think they have something a little different for 7th and 8th grade, but many schools use other textbooks. Usually, only the top level 8th grade math class is adequate. In our town, the public schools only offer "algebra lite" in 8th grade.

You state:Students will remember incorrect "discoveries" just as well as correct ones., but doesn't that depend on how well the "discovery" is reinforced? Won't the celebration of the discovery tend to reinforce long term memory retention? I think the greater problem is that there isn't sufficient practice to gain RESA for discovered facts.

To the other posters, I've been using FASTT Math with our son for him to master math facts. Getting fluency on +,-,x,/ is taking a lot more hours than I expected.

---Students will remember incorrect "discoveries" just as well as correct ones.

Of course, and this is a fundamental point. It's the reason that students make the same mistakes on tests that they made on homework-even after that homework is corrected. This repetition of mistakes occurs at any age, at any level or stage of learning before mastery.

It happens because a student uses what he has in long term and working memory to reason about how to solve a problem. They reach an endpoint. Depending on the problem, they may or may not recognize they were in error. After the error becomes clear (or is pointed out to them), they are not really any better off: they only know how to reason to the point of the error. Telling them "that's wrong, you should solve the problem by doing A---Z" doesn't help. Why? Because the LTM and WM didn't support doing A--Z in the first place, and you've not given them enough information to either see where they went off the rails, or to correct their misunderstanding. It's not enough to tell someone "solve the problem differently", you have to make them really understand why NOT to do what they did before. So until you've ground into them the right way, AND then showed them how to recognize why the wrong way is wrong, they will continue to make the same errors.

Constructivism leads people to the same errors, same stalemates and same dead ends. The frustration and pain that this causes is even more destructive to the learning process than the general confusion in the first place.

"Won't the celebration of the discovery tend to reinforce long term memory retention?"

Perhaps, but there are lots of facts and skills to learn, so you really can't depend on the light bulb effect as a regular tool.

"Getting fluency on +,-,x,/ is taking a lot more hours than I expected."

And I am finding that just because my son masters (?) them at one point in time, certain facts (e.g. 7 times 8) tend to slip away. I find simple flash cards quite effective and fast.

"The frustration and pain that this causes is even more destructive to the learning process than the general confusion in the first place."

It's a very inefficient way of learning on a consistent basis - and neither necessary or sufficient. As this comment states, it can also go very wrong. Discovery is done in groups so only one student gets the light bulb discovery moment. He/She then proceeds to "teach" it (right or wrong) to the other students. This is supposed to be better than having a well-trained teacher do the job? When the teacher gets all of the kids back together to talk about what they discovered, how is the teacher going to fix up the mess, even those who discovered a solution that works only for that one special case? There should be a reason why teachers go to college, and it's not about making learning more difficult, or creating problems.

I think their goal is not constructivism, but child-centered group learning with the teacher as guide on the side. The traditional teaching model is to have the teacher lead the whole class along a carefully-defined path that maximizes learning and minimizes wasted time. The teacher should know the material and should know all of the variations of mistakes and misunderstandings that kids make. There is no reason (I have done it myself) why a teacher cannot carefully lead kids along a path to discovery in class. This is a common teaching technique.

There is nothing about constructivism that requires child-centered groups that start with almost zero prior knowledge. Constructivism is also best left as an individually attempted homework assignment.

Schools would rather drop constructivism than lose their mixed-ability, child-centered learning groups.

"Also, I wonder is this something I am going to have to deal with year after year."

"Yes. Join the club."

And it's a big club.

If you do nothing but go to Superkids.com and print out math worksheets to get speed and fluency on math facts, you'll be doing quite a bit. I assure you that teachers do the same with their kids.

It isn't as hard as it seems once you settle in on the notion that you are going to have to protect your kid from this. I used to take 15 minutes in the morning to do "mad minute" sheets with both of my kids.

I tutored one son every day right after dinner for a half an hour to an hour for the last year and a half and he's come up a couple of years in math.

If you have no time and a little money, look into your nearest Kumon center.

If your grade school is determined to stick with EM (or any of the others are are similar), than you'll need to supplement all the way through. Long division won't be taught at all unless your teacher supplements, even though they'll need it for middle school and algebra.

It really isn't too hard once you settle into a routine. You just have to have a plan. If you need more than the sheets, Singapore math is inexpensive and easy to use. A page or two a day can really help.

---This repetition of mistakes occurs at any age, at any level or stage of learning before mastery.

As my husband and I sit with our eight-year-old son doing math homework, we notice him making the same mistakes consistently. When we try to correct him, he is convinced his answer is right. He says "but the teacher said this is how you do it." Did she say that or did he come to his own conclusion his answer was right after she did not correct him? Who knows?

How does a teacher measure mastery of a subject (particularly math) when there are no letter grades? Is it subjective? Does a teacher "feel" the student has mastered a math concept because he understands the concepts behind the math and not just the method? Our school does not have progress reports. Our report cards are numbers only. A two meaning developing, but needs further experience.

"Discovery is done in groups so only one student gets the light bulb discovery moment. He/She then proceeds to "teach" it (right or wrong) to the other students. This is supposed to be better than having a well-trained teacher do the job?"

I have asked myself that same question. Why bother requiring teachers to go to ed school if pupils can teach just as well or even better?

A consequence of the dominant ed school creed (constructivism, inquiry, discovery, minimal guidance...) is that teachers are not required to master subject matter since teachers are not supposed to impart knowledge and provide explicit instruction. Under the preferred form of non-instruction a teacher's lack of subject matter expertise never becomes an issue. So some pupils may indeed know more than the teacher.

The whole point of requiring teachers to go to ed school is to indoctrinate teachers in the ways of this idiotic anti-knowledge ed ideology.

The first requirement is that she have the addition and multiplication tables down cold. That is, I see

3(2x+6)-3

and I can immediately simplify it to

6x+15

That involves a simple use of a couple of higher order concepts (distribution and order of operations), but mainly it requires calculating 3x2=6 and 3x6-3=15 automatically. If you've got to stop and think about these calculations, you'll lose your place in the bigger problem.

So, drill and kill. If the school isn't doing it, assign some homework yourself.

About D-Ed Reckoning

The primary problem with K-12 education today is the problem of dead reckoning--an estimate based on little or no information. We don't know what a good K-12 education system is because we've never seen one operating. A good education system is one that is capable of educating almost every child.